Megagon

Regular megagon
Circle - black simple.svg
A regular megagon
Type Regular polygon
Edges and vertices 1000000
Schläfli symbol {1000000}
t{500000}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node 1.png
Symmetry group Dihedral (D1000000), order 2×1000000
Internal angle (degrees) 179.99964°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

A megagon is a polygon with 1 million sides (mega-, from the Greek μέγας megas, meaning "great").[1][2] Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.

A regular megagon is represented by Schläfli symbol {1000000} and can be constructed as a quasiregular truncated 500000-gon, t{500000}, which alternates two types of edges.

Properties

A regular megagon has an interior angle of 179.99964°.[1] The area of a regular megagon with sides of length a is given by

A = 250000a^2 \cot \frac{\pi}{1000000}.

The perimeter of a regular megagon inscribed in the unit circle is:

2000000 \sin\frac{\pi}{1000000},

which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be about 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.[3]

Because 1000000 = 26 × 56, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular megagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a power of two, three, or six.

Philosophical application

Like René Descartes' example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.[4][5][6][7][8][9][10]

The megagon is also used as an illustration of the convergence of regular polygons to a circle.[11]

Megagram

A megagram is an million-sided star polygon. There are 199,999 regular forms[12] given by Schläfli symbols of the form {1000000/n}, where n is an integer between 2 and 500,000 that is coprime to 1,000,000. There are also 300,000 regular star figures in the remaining cases.

References

  1. ^ a b Darling, David J., The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. Page 249. ISBN 0-471-27047-4.
  2. ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.
  3. ^ Williamson, Benjamin, An Elementary Treatise on the Differential Calculus, Longmans, Green, and Co., 1899. Page 45.
  4. ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
  5. ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.
  6. ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.
  7. ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.
  8. ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.
  9. ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
  10. ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.
  11. ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.
  12. ^ 199,999 = 500,000 cases - 1 (convex) - 100,000 (multiples of 5) - 250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)